Curvature-free effects from volume growth and ends-counting and their applications

TL;DR

This paper explores curvature-free geometric effects related to volume growth and ends-counting, providing new proofs and extensions of classical theorems without relying on traditional curvature comparison tools.

Contribution

It establishes new curvature-free theorems linking volume growth and ends-counting to geometric properties, extending classical results to broader curvature settings.

Findings

Complete manifolds with sublinear volume growth admit bounded mean-concave exhaustions.

Manifolds with infinitely many ends contain escaping geodesic lines outside compact sets.

New proofs of Calabi--Yau and Cai--Li--Tam theorems without Ricci curvature tools.

Abstract

In this paper, we investigate two curvature-free effects from volume growth and ends-counting, respectively. Motivated by generalizing classical results from Ricci curvature to other common curvatures, we establish two main theorems. First, any complete non-compact manifold with lower sublinear volume growth admits a smooth bounded mean-concave exhaustion. Second, any complete manifold with infinitely many ends contains escaping geodesic lines outside every compact subset. As applications, we provide new proofs of the Calabi--Yau minimal volume growth theorem and the Cai--Li--Tam finite-ends theorem for nonnegative Ricci curvature, without relying on the Bishop--Gromov volume comparison theorem or analytic tools specific to Ricci curvature. We further extend these results to Riemannian manifolds with nonnegative scalar curvature and K\"ahler manifolds with positive holomorphic sectional curvature.